Conventional DFT detectors known in the art are widely utilized in digital signal processing and test equipment to measure the spectra of signals from various sources and devices under test (DUTs). Such DFT detectors calculate the dot product of a sampled and digitized signal vector and the test vector of sampled complex-valued function Cos(2πft)−i Sin(2πft) of time t and test frequency f internally generated so that the test vector contains integer number of function cycles. This complex-valued test vector is also called test phasor. The dot product is a single complex value, the real part of which is often referred to as the in-phase component, and the imaginary as the quadrature component of the signal. The detector outputs holding these two values are referred to as the in-phase and quadrature channels of the detector.
To produce a spectrum DFT detectors generate a comb of test frequencies f within a range. The narrower the spacing between neighboring frequencies in the comb the higher the spectral resolution of the instrument. Due to the periodic nature of the test function however high spectral resolution requires very long test vectors to keep the number of cycles integral, which in practice leads to very long memory buffers and requires substantial computational power. While manageable in expensive test equipment, these factors are challenging in low-power remote monitoring systems and low-cost single-chip implementations.
It is also known in the art that while the test vector contains an integer number of cycles at all test frequencies within the spectrum, the frequency of input signal may not coincide with any of those test frequencies. In this case the vector of sampled signal does not contain the integer number of signal cycles—the signal is called “non-coherently sampled.” It means that the first and the last sample of the signal sinusoid are discontinuous with one another and the sampled signal vector is sometimes called “discontinuous.” This causes a problem well known in the art as “spectral leakage”: instead of producing a single peak at a signal frequency the components of such signal appear at a number of neighboring test frequencies creating erroneous broadening of spectral lines, false peaks and troughs, and a general increase in noise floor. The signal is said to be “leaking” from the frequency point where the signal peak is supposed to be located into the neighboring frequencies, where the test vectors are continuous.
It is also known in the art that DFT detectors can operate as a part of a network analyzer by synthesizing stimulus signal coherent with the continuous test vector or a component thereof, typically the Cos(2πft), which ensures stimulus continuity. The stimulus is applied to the device under test (DUT) and the device response is sampled as input signal to the detector. Except for special cases, this guarantees that the DUT's response is also continuous and the effects of spectral leakage are nonexistent.
What is overlooked in the art is that low-cost implementations of the DFT detectors, while formally calculating the same dot product of sampled signal vector and test vector, utilize a test vector of a limited length, which rarely contains an integer number of cycles and therefore is discontinuous. Such discontinuous test vectors do not constitute an orthogonal Fourier basis and therefore the calculated dot product should not be called “Fourier transform” and such a detector cannot be considered a true DFT detector.
The use of these discontinuous test vectors gives rise to a variety of unexpected artifacts in the detector output. Even in network analyzer mode, when the stimulus is coherently synthesized from a discontinuous test vector and applied to the DUT, the response is also discontinuous and the accuracy expected from a true DFT detector is not achieved. In the cited art, such deviations from expected DFT behavior are erroneously attributed to spectral leakage, when, in fact, they are completely different in nature.
It is well known in the art that if a true DFT detector is presented with a constant signal (DC), both the in-phase and quadrature output components are zero. It is also well known in the art that a constant signal cannot produce spectral leakage. If the DFT detector with discontinuous test vectors is presented with a constant DC signal, both the in-phase and quadrature output components show substantial frequency-dependent values, so the DC signal “leaks” into both detector outputs. For lack of established terminology, this phenomenon can be called “DC leakage.”
It is well known in the art that if an AC signal is coherently synthesized from the cosine component of a continuous test vector and delivered to the input of the DFT detector, the in-phase component will contain the amplitude of the synthesized signal, while the quadrature component will be zero and there will be no spectral leakage. If the same experiment is performed with a discontinuous test vector, the in-phase component will contain the distorted amplitude of the signal, while the quadrature component will be non-zero; therefore, some of the signal from the in-phase channel “leaks” into the quadrature channel and vice versa (intra-channel cross-talk). For lack of established terminology, this phenomenon can be called “AC leakage.”
The effects of the DC and AC leakage can exceed the measured signal when only a few full cycles and a fraction thereof fits within the length of the discontinuous test vector rendering the detector useless. The higher the test frequency, the more full cycles can fit within of the discontinuous test vector, the lower the DC and AC leakage. As test frequency increases, the levels of the DC and AC leakage decrease and may become lower than the acceptable error threshold for a given application; however, without the knowledge of the existence and behavior of DC and AC leakage, it is impossible to determine the frequency range in which a particular implementation of the detector can operate.
As the effects of the DC and AC leakage were erroneously attributed to spectral leakage, windowing—the well-known method for spectral leakage suppression—is applied. While helping to a certain degree, windowing further obscures the nature of the DC and AC leakage and still does not allow DFT detectors to operate at full frequency range and accuracy afforded by the hardware. The equations in the remaining text assume that Hann window is implemented in the DFT detector.
The present invention discloses:                i. the existence of the artifacts caused by discontinuous test vector: DC and AC leakage        ii. that DC and AC leakage are functions of the test vector length (i.e. number of samples) and the ratio of the test frequency to the sampling frequency        iii. that these functions can be expressed in close form for a practical implementation of the detector        iv. the way these functions are used to eliminate the DC and AC leakage from the detector output greatly increasing the detector accuracy, expanding operation frequency range and without any additional hardware        
While it is known in the art that conventional DFT detectors are blind to DC signals, the present invention discloses the use of the DC leakage to enable measurement of DC signals by DFT detectors with discontinuous test vector.
While there may be multiple ways of implementing DFT detectors utilizing various hardware designs and digital signal processors (DSPs), of a particular interest for practical applications are the commercially available, single-chip DFT devices designed specifically for impedimetric applications by Analog Devices:    (32) AD5933 Datasheet; Analog Devices: Norwood, Mass., USA, 2007.    (33) Caffrey, J. F.; Geraghty, D. P.; Lyden, C. G.; O'Grady, A. C.; Slattery, C. F.; Smith, S. Measuring Circuit and a Method for Determining a Characteristic of the Impedance of a Complex Impedance Element for Facilitating Characterization of the Impedance Thereof. U.S. Pat. No. 7,555,394, June, 2009.    (34) Evaluation Board for the 1 Msps 12-bit Impedance Converter Network Analyzer; Analog Devices Technical Report; Analog Devices, Norwood, Mass., USA, 2005.    (35) Leonard, E. Optimize Speaker Impedance Matching for Best Audio Results. EE Times, 26 Apr. 2006, p. id=1274771.    (36) Brennan, S. Measuring a Loudspeaker Impedance Profile Using the AD5933; Analog Devices Application Note AN-843; Analog Devices, Norwood, Mass., USA, 2007; pp. 1-12.    (37) ADuCM350 Datasheet; Analog Devices: Norwood, Mass., USA, 2014.
All the art cited above is predominantly focusing on the end applications and lacks specific error analysis in conjunction with operation frequency range and calibration methods, relying heavily on the information from the device datasheet. For example, the operating frequency range with a 16 MHz clock is stated to be from 1 KHz to 100 KHz with the system accuracy of 0.5%. for the impedance dynamic range of 1 kΩ to 10 MΩ, but no experimental data on accuracy given.
The calibration procedure proposed in the AD5933 datasheet and widely replicated in the literature does not take into account the artifacts caused by discontinuous test vector (phasor): resulting from the DC offset at the detector input and the cross-talk between in-phase and quadrature channels. The calibration procedure described in the art produces a single multiplicative gain factor that leads to rather sizeable systematic errors and undue disappointment in the device performance (see publication (4)), especially at the lower end of the operation frequency range.
Notwithstanding the various advances known in the art in connection with utilization of DFT detectors, there remains a need in the art for improvements, especially improvements which enhance the accuracy and versatility of such detectors in a variety of applications, allowing for reduction in hardware, footprint, power consumption, cost and environmental impact of the resulting electronic products.